
for \( \beta = 1,2,4 \) .
If \( \beta = 2\) , then SDE (REF ) describes the dynamics of the eigenvalues of Gaussian unitary ensembles of order \( N\in \mathbb {N}\)  [1]}, [2]}.
By taking \( \beta = 2 \)  and letting \( N\rightarrow \infty \)  in Eq. (REF ),
Spohn introduced the infinite-dimensional stochastic differential equation (ISDE)
\(& X_t^i - X_0^i = B_t^i + \int _0^t \sum _{ j\ne  i , \, j \in \mathbb {Z}}\frac{1}{X_u^i - X_u^j } du \quad (i\in \mathbb {Z}).\) 
